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Motivated by the construction of Bach flat neutral signature Riemannian extensions, we study the space of parallel trace-free tensors of type (1, 1) on an affine surface. It is shown that the existence of such a parallel tensor field is characterized by the recurrence of the symmetric part of the Ricci tensor.

Affine Surfaces Which are Kähler, Para-Kähler, or Nilpotent Kähler